§26.8 Set Partitions: Stirling Numbers

Keywords:: partitions
Referenced by:: §26.13, §26.14(iii)
Permalink:: http://dlmf.nist.gov/26.8

§26.8(i) Definitions
§26.8(ii) Generating Functions
§26.8(iii) Special Values
§26.8(iv) Recurrence Relations
§26.8(v) Identities
§26.8(vi) Relations to Bernoulli Numbers
§26.8(vii) Asymptotic Approximations

§26.8(i) Definitions

Notes:: See Comtet (1974, pp. 206–207, 214), Riordan (1979, p. 195). Tables 26.8.1 and 26.8.2 are from Abramowitz and Stegun (1964, Tables 24.3 and 24.4).
Defines:: $\mathop{s\/}\nolimits\!\left(n,k\right)$ : Stirling number of the first kind and $\mathop{S\/}\nolimits\!\left(n,k\right)$ : Stirling number of the second kind
Keywords:: Stirling numbers (first and second kinds), partitions
Referenced by:: §24.15(iii), §26.17, §26.7(i)
Permalink:: http://dlmf.nist.gov/26.8.i

$\mathop{s\/}\nolimits\!\left(n,k\right)$ denotes the Stirling number of the first kind: $(-1)^{{n-k}}$ times the number of permutations of $\{ 1,2,\ldots,n\}$ with exactly $k$ cycles. See Table 26.8.1.

26.8.1 $\mathop{s\/}\nolimits\!\left(n,n\right)=1,$ $n\geq 0$ ,

Symbols:: $\mathop{s\/}\nolimits\!\left(n,k\right)$ : Stirling number of the first kind and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/26.8.E1
Encodings:: TeX, pMML, png

26.8.2 $\mathop{s\/}\nolimits\!\left(1,k\right)=\delta _{{1,k}},$

Symbols:: $\delta _{{j,k}}$ : Kronecker delta, $\mathop{s\/}\nolimits\!\left(n,k\right)$ : Stirling number of the first kind and $k$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/26.8.E2
Encodings:: TeX, pMML, png

26.8.3 $(-1)^{{n-k}}\mathop{s\/}\nolimits\!\left(n,k\right)=\sum _{{1\leq b_{1}<\dots<b_{{n-k}}\leq n-1}}b_{1}b_{2}\cdots b_{{n-k}},$ $n>k\geq 1$ .

Symbols:: $\mathop{s\/}\nolimits\!\left(n,k\right)$ : Stirling number of the first kind, $k$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/26.8.E3
Encodings:: TeX, pMML, png

$\mathop{S\/}\nolimits\!\left(n,k\right)$ denotes the Stirling number of the second kind: the number of partitions of $\{ 1,2,\ldots,n\}$ into exactly $k$ nonempty subsets. See Table 26.8.2.

26.8.4 $\mathop{S\/}\nolimits\!\left(n,n\right)=1,$ $n\geq 0$ ,

Symbols:: $\mathop{S\/}\nolimits\!\left(n,k\right)$ : Stirling number of the second kind and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/26.8.E4
Encodings:: TeX, pMML, png

26.8.5 $\mathop{S\/}\nolimits\!\left(n,k\right)=\sum 1^{{c_{1}}}2^{{c_{2}}}\cdots k^{{c_{k}}},$

Symbols:: $\mathop{S\/}\nolimits\!\left(n,k\right)$ : Stirling number of the second kind, $k$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/26.8.E5
Encodings:: TeX, pMML, png

where the summation is over all nonnegative integers $c_{1},c_{2},\ldots,c_{k}$ such that $c_{1}+c_{2}+\cdots+c_{k}=n-k.$

26.8.6 $\mathop{S\/}\nolimits\!\left(n,k\right)=\frac{1}{k!}\sum _{{j=0}}^{k}(-1)^{{k-j}}\binom{k}{j}j^{n}.$

Symbols:: $\mathop{S\/}\nolimits\!\left(n,k\right)$ : Stirling number of the second kind, $\binom{m}{n}$ : binomial coefficient, $!$ : $n!$ : factorial, $j$ : nonnegative integer, $k$ : nonnegative integer and $n$ : nonnegative integer
A&S Ref:: 24.1.4
Permalink:: http://dlmf.nist.gov/26.8.E6
Encodings:: TeX, pMML, png

Table 26.8.1: Stirling numbers of the first kind $\mathop{s\/}\nolimits\!\left(n,k\right)$ .

$n$	$k$
$n$	0	1	2	3	4	5	6	7	8	9	10
0	1
1	0	1
2	0	−1	1
3	0	2	−3	1
4	0	−6	11	−6	1
5	0	24	−50	35	−10	1
6	0	−120	274	−225	85	−15	1
7	0	720	−1764	1624	−735	175	−21	1
8	0	−5040	13068	−13132	6769	−1960	322	−28	1
9	0	40320	−1 09584	1 18124	−67284	22449	−4536	546	−36	1
10	0	−3 62880	10 26576	−11 72700	7 23680	−2 69325	6327	−9450	870	−45	1

Symbols:: $\mathop{s\/}\nolimits\!\left(n,k\right)$ : Stirling number of the first kind, $k$ : nonnegative integer and $n$ : nonnegative integer
Keywords:: Stirling numbers (first and second kinds)
Referenced by:: §26.8(i), §26.8(i)
Permalink:: http://dlmf.nist.gov/26.8.T1

Table 26.8.2: Stirling numbers of the second kind $\mathop{S\/}\nolimits\!\left(n,k\right)$ .

$n$	$k$
$n$	0	1	2	3	4	5	6	7	8	9	10
0	1
1	0	1
2	0	1	1
3	0	1	3	1
4	0	1	7	6	1
5	0	1	15	25	10	1
6	0	1	31	90	65	15	1
7	0	1	63	301	350	140	21	1
8	0	1	127	966	1701	1050	266	28	1
9	0	1	255	3025	7770	6951	2646	462	36	1
10	0	1	511	9330	34105	42525	22827	5880	750	45	1

Symbols:: $\mathop{S\/}\nolimits\!\left(n,k\right)$ : Stirling number of the second kind, $k$ : nonnegative integer and $n$ : nonnegative integer
Keywords:: Stirling numbers (first and second kinds)
Referenced by:: §26.8(i), §26.8(i)
Permalink:: http://dlmf.nist.gov/26.8.T2

§26.8(ii) Generating Functions

Notes:: See Comtet (1974, pp. 206–213).
Keywords:: Stirling numbers (first and second kinds)
Permalink:: http://dlmf.nist.gov/26.8.ii

26.8.7 $\sum _{{k=0}}^{n}\mathop{s\/}\nolimits\!\left(n,k\right)x^{k}=(x-n+1)_{n},$

Symbols:: $\mathop{s\/}\nolimits\!\left(n,k\right)$ : Stirling number of the first kind, $x$ : real variable, $k$ : nonnegative integer and $n$ : nonnegative integer
A&S Ref:: 24.1.3 (in slightly different form)
Permalink:: http://dlmf.nist.gov/26.8.E7
Encodings:: TeX, pMML, png

where $\left(x\right)_{{n}}$ is the Pochhammer symbol: $x(x+1)\cdots(x+n-1)$ .

26.8.8 $\sum _{{n=0}}^{{\infty}}\mathop{s\/}\nolimits\!\left(n,k\right)\frac{x^{n}}{n!}=\frac{(\mathop{\ln\/}\nolimits(1+x))^{k}}{k!},$ $|x|<1$ ,

Symbols:: $\mathop{s\/}\nolimits\!\left(n,k\right)$ : Stirling number of the first kind, $!$ : $n!$ : factorial, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $x$ : real variable, $k$ : nonnegative integer and $n$ : nonnegative integer
A&S Ref:: 24.1.3
Permalink:: http://dlmf.nist.gov/26.8.E8
Encodings:: TeX, pMML, png

26.8.9 $\sum _{{n,k=0}}^{{\infty}}\mathop{s\/}\nolimits\!\left(n,k\right)\frac{x^{n}}{n!}y^{k}=(1+x)^{y},$ $|x|<1$ .

Symbols:: $\mathop{s\/}\nolimits\!\left(n,k\right)$ : Stirling number of the first kind, $!$ : $n!$ : factorial, $x$ : real variable, $k$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/26.8.E9
Encodings:: TeX, pMML, png

26.8.10 $\sum _{{k=1}}^{n}\mathop{S\/}\nolimits\!\left(n,k\right)(x-k+1)_{k}=x^{n},$

Symbols:: $\mathop{S\/}\nolimits\!\left(n,k\right)$ : Stirling number of the second kind, $x$ : real variable, $k$ : nonnegative integer and $n$ : nonnegative integer
A&S Ref:: 24.1.4
Referenced by:: ¶ ‣ §26.15
Permalink:: http://dlmf.nist.gov/26.8.E10
Encodings:: TeX, pMML, png

26.8.11 $\sum _{{n=0}}^{{\infty}}\mathop{S\/}\nolimits\!\left(n,k\right)\, x^{n}=\frac{x^{k}}{(1-x)(1-2x)\cdots(1-kx)},$ $|x|<1/k$ ,

Symbols:: $\mathop{S\/}\nolimits\!\left(n,k\right)$ : Stirling number of the second kind, $x$ : real variable, $k$ : nonnegative integer and $n$ : nonnegative integer
A&S Ref:: 24.1.4
Permalink:: http://dlmf.nist.gov/26.8.E11
Encodings:: TeX, pMML, png

26.8.12 $\sum _{{n=0}}^{{\infty}}\mathop{S\/}\nolimits\!\left(n,k\right)\frac{x^{n}}{n!}=\frac{(e^{x}-1)^{k}}{k!},$

Symbols:: $\mathop{S\/}\nolimits\!\left(n,k\right)$ : Stirling number of the second kind, $e$ : base of exponential function, $!$ : $n!$ : factorial, $x$ : real variable, $k$ : nonnegative integer and $n$ : nonnegative integer
A&S Ref:: 24.1.4
Permalink:: http://dlmf.nist.gov/26.8.E12
Encodings:: TeX, pMML, png

26.8.13 $\sum _{{n,k=0}}^{{\infty}}\mathop{S\/}\nolimits\!\left(n,k\right)\frac{x^{n}}{n!}y^{k}=\mathop{\exp\/}\nolimits\left(y(e^{x}-1)\right).$

Symbols:: $\mathop{S\/}\nolimits\!\left(n,k\right)$ : Stirling number of the second kind, $\mathop{\exp\/}\nolimits z$ : exponential function, $e$ : base of exponential function, $!$ : $n!$ : factorial, $x$ : real variable, $k$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/26.8.E13
Encodings:: TeX, pMML, png

§26.8(iii) Special Values

Notes:: See Graham et al. (1994, p. 264).
Keywords:: Stirling numbers (first and second kinds)
Permalink:: http://dlmf.nist.gov/26.8.iii

For $n\geq 1$ ,

26.8.14

$\mathop{s\/}\nolimits\!\left(n,0\right)=0,$

$\mathop{s\/}\nolimits\!\left(n,1\right)=(-1)^{{n-1}}(n-1)!,$

Symbols:: $\mathop{s\/}\nolimits\!\left(n,k\right)$ : Stirling number of the first kind, $!$ : $n!$ : factorial and $n$ : nonnegative integer
A&S Ref:: 24.1.3
Permalink:: http://dlmf.nist.gov/26.8.E14
Encodings:: TeX, TeX, pMML, pMML, png, png

26.8.15 $\mathop{s\/}\nolimits\!\left(n,2\right)=(-1)^{n}(n-1)!\left(1+\frac{1}{2}+\cdots+\frac{1}{n-1}\right),$

Symbols:: $\mathop{s\/}\nolimits\!\left(n,k\right)$ : Stirling number of the first kind, $!$ : $n!$ : factorial and $n$ : nonnegative integer
A&S Ref:: 24.1.3
Permalink:: http://dlmf.nist.gov/26.8.E15
Encodings:: TeX, pMML, png

26.8.16 $-\mathop{s\/}\nolimits\!\left(n,n-1\right)=\mathop{S\/}\nolimits\!\left(n,n-1\right)=\binom{n}{2},$

Symbols:: $\mathop{s\/}\nolimits\!\left(n,k\right)$ : Stirling number of the first kind, $\mathop{S\/}\nolimits\!\left(n,k\right)$ : Stirling number of the second kind, $\binom{m}{n}$ : binomial coefficient and $n$ : nonnegative integer
A&S Ref:: 24.1.3
Permalink:: http://dlmf.nist.gov/26.8.E16
Encodings:: TeX, pMML, png

26.8.17

$\mathop{S\/}\nolimits\!\left(n,0\right)=0,$

$\mathop{S\/}\nolimits\!\left(n,1\right)=1,$

$\mathop{S\/}\nolimits\!\left(n,2\right)=2^{{n-1}}-1.$

Symbols:: $\mathop{S\/}\nolimits\!\left(n,k\right)$ : Stirling number of the second kind and $n$ : nonnegative integer
A&S Ref:: 24.1.4
Permalink:: http://dlmf.nist.gov/26.8.E17
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png

§26.8(iv) Recurrence Relations

Notes:: See Comtet (1974, pp. 208–214), Graham et al. (1994, p. 265), Riordan (1979, pp. 204–227).
Keywords:: Stirling numbers (first and second kinds)
Permalink:: http://dlmf.nist.gov/26.8.iv

26.8.18 $\mathop{s\/}\nolimits\!\left(n,k\right)=\mathop{s\/}\nolimits\!\left(n-1,k-1\right)-(n-1)\mathop{s\/}\nolimits\!\left(n-1,k\right),$

Symbols:: $\mathop{s\/}\nolimits\!\left(n,k\right)$ : Stirling number of the first kind, $k$ : nonnegative integer and $n$ : nonnegative integer
A&S Ref:: 24.1.3
Permalink:: http://dlmf.nist.gov/26.8.E18
Encodings:: TeX, pMML, png

26.8.19 $\binom{k}{h}\mathop{s\/}\nolimits\!\left(n,k\right)=\sum _{{j=k-h}}^{{n-h}}\binom{n}{j}\mathop{s\/}\nolimits\!\left(n-j,h\right)\mathop{s\/}\nolimits\!\left(j,k-h\right),$ $n\geq k\geq h$ ,

Symbols:: $\mathop{s\/}\nolimits\!\left(n,k\right)$ : Stirling number of the first kind, $\binom{m}{n}$ : binomial coefficient, $h$ : nonnegative integer, $j$ : nonnegative integer, $k$ : nonnegative integer and $n$ : nonnegative integer
A&S Ref:: 24.1.3
Permalink:: http://dlmf.nist.gov/26.8.E19
Encodings:: TeX, pMML, png

26.8.20 $\mathop{s\/}\nolimits\!\left(n+1,k+1\right)=n!\sum _{{j=k}}^{{n}}\frac{(-1)^{{n-j}}}{j!}\,\mathop{s\/}\nolimits\!\left(j,k\right),$

Symbols:: $\mathop{s\/}\nolimits\!\left(n,k\right)$ : Stirling number of the first kind, $!$ : $n!$ : factorial, $j$ : nonnegative integer, $k$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/26.8.E20
Encodings:: TeX, pMML, png

26.8.21 $\mathop{s\/}\nolimits\!\left(n+k+1,k\right)=-\sum _{{j=0}}^{{k}}(n+j)\mathop{s\/}\nolimits\!\left(n+j,j\right).$

Symbols:: $\mathop{s\/}\nolimits\!\left(n,k\right)$ : Stirling number of the first kind, $j$ : nonnegative integer, $k$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/26.8.E21
Encodings:: TeX, pMML, png

26.8.22 $\mathop{S\/}\nolimits\!\left(n,k\right)=k\mathop{S\/}\nolimits\!\left(n-1,k\right)+\mathop{S\/}\nolimits\!\left(n-1,k-1\right),$

Symbols:: $\mathop{S\/}\nolimits\!\left(n,k\right)$ : Stirling number of the second kind, $k$ : nonnegative integer and $n$ : nonnegative integer
A&S Ref:: 24.1.4
Permalink:: http://dlmf.nist.gov/26.8.E22
Encodings:: TeX, pMML, png

26.8.23 $\binom{k}{h}\mathop{S\/}\nolimits\!\left(n,k\right)=\sum _{{j=k-h}}^{{n-h}}\binom{n}{j}\mathop{S\/}\nolimits\!\left(n-j,h\right)\mathop{S\/}\nolimits\!\left(j,k-h\right),$ $n\geq k\geq h$ ,

Symbols:: $\mathop{S\/}\nolimits\!\left(n,k\right)$ : Stirling number of the second kind, $\binom{m}{n}$ : binomial coefficient, $h$ : nonnegative integer, $j$ : nonnegative integer, $k$ : nonnegative integer and $n$ : nonnegative integer
A&S Ref:: 24.1.4
Permalink:: http://dlmf.nist.gov/26.8.E23
Encodings:: TeX, pMML, png

26.8.24 $\mathop{S\/}\nolimits\!\left(n,k\right)=\sum _{{j=k}}^{n}\mathop{S\/}\nolimits\!\left(j-1,k-1\right)k^{{n-j}},$

Symbols:: $\mathop{S\/}\nolimits\!\left(n,k\right)$ : Stirling number of the second kind, $j$ : nonnegative integer, $k$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/26.8.E24
Encodings:: TeX, pMML, png

26.8.25 $\mathop{S\/}\nolimits\!\left(n+1,k+1\right)=\sum _{{j=k}}^{{n}}\binom{n}{j}\mathop{S\/}\nolimits\!\left(j,k\right),$

Symbols:: $\mathop{S\/}\nolimits\!\left(n,k\right)$ : Stirling number of the second kind, $\binom{m}{n}$ : binomial coefficient, $j$ : nonnegative integer, $k$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/26.8.E25
Encodings:: TeX, pMML, png

26.8.26 $\mathop{S\/}\nolimits\!\left(n+k+1,k\right)=\sum _{{j=0}}^{{k}}j\mathop{S\/}\nolimits\!\left(n+j,j\right).$

Symbols:: $\mathop{S\/}\nolimits\!\left(n,k\right)$ : Stirling number of the second kind, $j$ : nonnegative integer, $k$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/26.8.E26
Encodings:: TeX, pMML, png

§26.8(v) Identities

Notes:: See Comtet (1974, pp. 213–216), Graham et al. (1994, pp. 264–265). Equation (26.8.37) is in Riordan (1979, p. 203). It and (26.8.39) imply (26.8.31). Equations (26.8.34) and (26.8.35) follow from equation (34) in Riordan (1979, p. 218).
Keywords:: Stirling numbers (first and second kinds)
Permalink:: http://dlmf.nist.gov/26.8.v

26.8.27 $\mathop{s\/}\nolimits\!\left(n,n-k\right)=\sum _{{j=0}}^{{k}}(-1)^{j}\binom{n-1+j}{k+j}\,\binom{n+k}{k-j}\*\mathop{S\/}\nolimits\!\left(k+j,j\right),$

Symbols:: $\mathop{s\/}\nolimits\!\left(n,k\right)$ : Stirling number of the first kind, $\mathop{S\/}\nolimits\!\left(n,k\right)$ : Stirling number of the second kind, $\binom{m}{n}$ : binomial coefficient, $j$ : nonnegative integer, $k$ : nonnegative integer and $n$ : nonnegative integer
A&S Ref:: 24.1.3
Permalink:: http://dlmf.nist.gov/26.8.E27
Encodings:: TeX, pMML, png

26.8.28 $\sum _{{k=1}}^{n}\mathop{s\/}\nolimits\!\left(n,k\right)=0,$ $n>1$ ,

Symbols:: $\mathop{s\/}\nolimits\!\left(n,k\right)$ : Stirling number of the first kind, $k$ : nonnegative integer and $n$ : nonnegative integer
A&S Ref:: 24.1.3
Permalink:: http://dlmf.nist.gov/26.8.E28
Encodings:: TeX, pMML, png

26.8.29 $\sum _{{k=1}}^{n}(-1)^{{n-k}}\mathop{s\/}\nolimits\!\left(n,k\right)=n!,$

Symbols:: $\mathop{s\/}\nolimits\!\left(n,k\right)$ : Stirling number of the first kind, $!$ : $n!$ : factorial, $k$ : nonnegative integer and $n$ : nonnegative integer
A&S Ref:: 24.1.3
Permalink:: http://dlmf.nist.gov/26.8.E29
Encodings:: TeX, pMML, png

26.8.30 $\sum _{{j=k}}^{n}\mathop{s\/}\nolimits\!\left(n+1,j+1\right)\, n^{{j-k}}=\mathop{s\/}\nolimits\!\left(n,k\right).$

Symbols:: $\mathop{s\/}\nolimits\!\left(n,k\right)$ : Stirling number of the first kind, $j$ : nonnegative integer, $k$ : nonnegative integer and $n$ : nonnegative integer
A&S Ref:: 24.1.3
Permalink:: http://dlmf.nist.gov/26.8.E30
Encodings:: TeX, pMML, png

26.8.31 $\frac{1}{k!}\,\frac{{d}^{k}}{{dx}^{k}}f(x)=\sum _{{n=k}}^{{\infty}}\frac{\mathop{s\/}\nolimits\!\left(n,k\right)}{n!}\,\Delta^{n}f(x),$

Symbols:: $\mathop{s\/}\nolimits\!\left(n,k\right)$ : Stirling number of the first kind, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $!$ : $n!$ : factorial, $\Delta$ : forward difference operator, $x$ : real variable, $k$ : nonnegative integer and $n$ : nonnegative integer
A&S Ref:: 24.1.3
Referenced by:: §26.8(v)
Permalink:: http://dlmf.nist.gov/26.8.E31
Encodings:: TeX, pMML, png

when $f(x)$ is analytic for all $x$ , and the series converges, where

26.8.32 $\Delta f(x)=f(x+1)-f(x);$

Symbols:: $\Delta$ : forward difference operator and $x$ : real variable
Permalink:: http://dlmf.nist.gov/26.8.E32
Encodings:: TeX, pMML, png

compare §3.6(i).

26.8.33 $\mathop{S\/}\nolimits\!\left(n,n-k\right)=\sum _{{j=0}}^{{k}}(-1)^{j}\binom{n-1+j}{k+j}\binom{n+k}{k-j}\*\mathop{s\/}\nolimits\!\left(k+j,j\right),$

Symbols:: $\mathop{s\/}\nolimits\!\left(n,k\right)$ : Stirling number of the first kind, $\mathop{S\/}\nolimits\!\left(n,k\right)$ : Stirling number of the second kind, $\binom{m}{n}$ : binomial coefficient, $j$ : nonnegative integer, $k$ : nonnegative integer and $n$ : nonnegative integer
A&S Ref:: 24.1.4
Permalink:: http://dlmf.nist.gov/26.8.E33
Encodings:: TeX, pMML, png

26.8.34 $\sum _{{j=0}}^{n}j^{k}x^{j}=\sum _{{j=0}}^{k}\mathop{S\/}\nolimits\!\left(k,j\right)x^{j}\frac{{d}^{j}}{{dx}^{j}}\left(\frac{1-x^{{n+1}}}{1-x}\right),$

Symbols:: $\mathop{S\/}\nolimits\!\left(n,k\right)$ : Stirling number of the second kind, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $x$ : real variable, $j$ : nonnegative integer, $k$ : nonnegative integer and $n$ : nonnegative integer
A&S Ref:: 24.1.4
Referenced by:: §26.8(v)
Permalink:: http://dlmf.nist.gov/26.8.E34
Encodings:: TeX, pMML, png

26.8.35 $\sum _{{j=0}}^{n}j^{k}=\sum _{{j=0}}^{k}j!\mathop{S\/}\nolimits\!\left(k,j\right)\binom{n+1}{j+1},$

Symbols:: $\mathop{S\/}\nolimits\!\left(n,k\right)$ : Stirling number of the second kind, $\binom{m}{n}$ : binomial coefficient, $!$ : $n!$ : factorial, $j$ : nonnegative integer, $k$ : nonnegative integer and $n$ : nonnegative integer
A&S Ref:: 24.1.4
Referenced by:: §26.8(v)
Permalink:: http://dlmf.nist.gov/26.8.E35
Encodings:: TeX, pMML, png

26.8.36 $\sum _{{k=0}}^{n}(-1)^{{n-k}}k!\mathop{S\/}\nolimits\!\left(n,k\right)=1.$

Symbols:: $\mathop{S\/}\nolimits\!\left(n,k\right)$ : Stirling number of the second kind, $!$ : $n!$ : factorial, $k$ : nonnegative integer and $n$ : nonnegative integer
A&S Ref:: 24.1.4
Permalink:: http://dlmf.nist.gov/26.8.E36
Encodings:: TeX, pMML, png

26.8.37 $\frac{1}{k!}\Delta^{k}f(x)=\sum _{{n=k}}^{{\infty}}\frac{\mathop{S\/}\nolimits\!\left(n,k\right)}{n!}\frac{{d}^{n}}{{dx}^{n}}f(x),$

Symbols:: $\mathop{S\/}\nolimits\!\left(n,k\right)$ : Stirling number of the second kind, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $!$ : $n!$ : factorial, $\Delta$ : forward difference operator, $x$ : real variable, $k$ : nonnegative integer and $n$ : nonnegative integer
A&S Ref:: 24.1.4
Referenced by:: §26.8(v)
Permalink:: http://dlmf.nist.gov/26.8.E37
Encodings:: TeX, pMML, png

when $f(x)$ is analytic for all $x$ , and the series converges.

Let $A$ and $B$ be the $n\times n$ matrices with $(j,k)$ th elements $\mathop{s\/}\nolimits\!\left(j,k\right)$ , and $\mathop{S\/}\nolimits\!\left(j,k\right)$ , respectively. Then

26.8.38 $A^{{-1}}=B.$

A&S Ref:: 24.1.4
Permalink:: http://dlmf.nist.gov/26.8.E38
Encodings:: TeX, pMML, png

26.8.39 $\sum _{{j=k}}^{n}\mathop{s\/}\nolimits\!\left(j,k\right)\mathop{S\/}\nolimits\!\left(n,j\right)=\sum _{{j=k}}^{n}\mathop{s\/}\nolimits\!\left(n,j\right)\mathop{S\/}\nolimits\!\left(j,k\right)=\delta _{{n,k}}.$

Symbols:: $\delta _{{j,k}}$ : Kronecker delta, $\mathop{s\/}\nolimits\!\left(n,k\right)$ : Stirling number of the first kind, $\mathop{S\/}\nolimits\!\left(n,k\right)$ : Stirling number of the second kind, $j$ : nonnegative integer, $k$ : nonnegative integer and $n$ : nonnegative integer
A&S Ref:: 24.1.4
Referenced by:: §26.8(v)
Permalink:: http://dlmf.nist.gov/26.8.E39
Encodings:: TeX, pMML, png

§26.8(vi) Relations to Bernoulli Numbers

Permalink:: http://dlmf.nist.gov/26.8.vi

See §24.15(iii).

§26.8(vii) Asymptotic Approximations

Notes:: See Jordan (1939, pp. 161–174) (as corrected here).
Keywords:: Stirling numbers (first and second kinds), partitions
Permalink:: http://dlmf.nist.gov/26.8.vii

26.8.40 $\mathop{s\/}\nolimits\!\left(n+1,k+1\right)\sim(-1)^{{n-k}}\frac{n!}{k!}(\EulerConstant+\mathop{\ln\/}\nolimits n)^{k},$ $n\to\infty$ ,

Symbols:: $\EulerConstant$ : Euler’s constant, $\mathop{s\/}\nolimits\!\left(n,k\right)$ : Stirling number of the first kind, $!$ : $n!$ : factorial, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\sim$ : asymptotic equality, $k$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/26.8.E40
Encodings:: TeX, pMML, png

uniformly for $k=\mathop{o\/}\nolimits\!\left(\mathop{\ln\/}\nolimits n\right)$ , where $\EulerConstant$ is Euler’s constant (§5.2(ii)).

26.8.41 $\mathop{s\/}\nolimits\!\left(n+k,k\right)\sim\frac{(-1)^{{n}}}{2^{n}n!}k^{{2n}},$ $k\to\infty$ ,

Symbols:: $\mathop{s\/}\nolimits\!\left(n,k\right)$ : Stirling number of the first kind, $!$ : $n!$ : factorial, $\sim$ : asymptotic equality, $k$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/26.8.E41
Encodings:: TeX, pMML, png

$n$ fixed.

26.8.42 $\mathop{S\/}\nolimits\!\left(n,k\right)\sim\frac{k^{n}}{k!},$ $n\to\infty$ ,

Symbols:: $\mathop{S\/}\nolimits\!\left(n,k\right)$ : Stirling number of the second kind, $!$ : $n!$ : factorial, $\sim$ : asymptotic equality, $k$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/26.8.E42
Encodings:: TeX, pMML, png

$k$ fixed.

26.8.43 $\mathop{S\/}\nolimits\!\left(n+k,k\right)\sim\frac{k^{{2n}}}{2^{n}n!},$ $k\to\infty$ ,

Symbols:: $\mathop{S\/}\nolimits\!\left(n,k\right)$ : Stirling number of the second kind, $!$ : $n!$ : factorial, $\sim$ : asymptotic equality, $k$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/26.8.E43
Encodings:: TeX, pMML, png

uniformly for $n=\mathop{o\/}\nolimits\!\left(k^{{1/2}}\right)$ .

For asymptotic approximations for $\mathop{s\/}\nolimits\!\left(n+1,k+1\right)$ and $\mathop{S\/}\nolimits\!\left(n,k\right)$ that apply uniformly for $1\leq k\leq n$ as $n\to\infty$ see Temme (1993).

For other asymptotic approximations and also expansions see Moser and Wyman (1958a) for Stirling numbers of the first kind, and Moser and Wyman (1958b), Bleick and Wang (1974) for Stirling numbers of the second kind.

For asymptotic estimates for generalized Stirling numbers see Chelluri et al. (2000).